College Algebra Week 3Chapter 3

The University of Phoenix

Inst. John Ensworth

Where to from here?

• We have played with equations.

• We have played with inequalities

• We have played with the form of a linear equation y= mx + b .

• Secretly the m and b were very special. m is called the slope and b is called the y intercept.

Let’s SEE what is happening

• One equation looks like the next until you make a picture

• “Why don’t you take a picture, it will last longer!”

So let’s build up our graphing skills

• First new definitions: Ordered Pairs

• They are the two numbers that make our special equation y = mx + b work.

• They are the answers that go together.

• For y = 2x -1 we can use (2,3) (where we always say (x,y) )

• 3=2(2)-1 3=3 Remember doing this?

More than one answer

• If you change either x or y, then the other number changes. You can have MANY answers (and all those answers together make the graph we’ll eventually get to).

Example 1 pg172

• Using y= -3x + 4

• Complete the ordered pair

• a) (2, ) y=-3*2+4 = -6+4 = -2 (2,-2)

• b) ( ,-5) -5 = -3x +4

-5-4 = -3x

-9 = -3x

3 = x (2,3)

Example 1 continued

• y=-3x+4

• c) (0, ) Remember, this is just (x,y)y= -3*0 +4

y=4 (0,4)

**Ex. 7-22**

The playing field

• The rectangularcoordinatesystem

• (also calledCartesian)

• Or even the coordinateplane

3

Parts is parts

• Important parts:

• The origin

• Quadrants

• X-axis

• Y-axis

Quick helpful tips

• When you sketch one, make sure your hash marks for the numbers are equally spaced on each and both the X-axis and the Y-axis

• You don’t have to plot the name of the points unless instructed to or if it is otherwise confusing

Plotting Points

• Just remember

(x,y)

Plot them!

• This is (2,3) top

• (-3,-2)

bottom

Example 2 pg174

• Plot the points (2,5), (-1,4), (-3,-4), (3,-2)

**Ex. 23-50**

Graphing an equation (a linear equation for this class)

Example 3 pg 175

• If you have y=2x-1, you can try many different x’s and see what y’s you get, then plot all of them…

• If x = -3 -2 -1 0 1 2 3

then

• y=2x-1= -7 -5 -3 -1 1 3 5

This gives us… plotted…• Each x gave us

a y from the equationy=2x-1

Then connectthe dots• And we

write in the equation totell later folkwhat is plotted

**Ex. 51-58**

Definition• The Linear Equation in Two Variables

• It is one written as Ax+By=C

• Where A,B,C are real numbers and both A and B cannot be = 0 at the same time.

A photo album of linear equations in two variables

• x-y=5

• y=2x+3

• 2x-5y-9=0

• x=8 (B = 0 this time)

• We can solve all these to look like y=mx+b and then graph them!

Example 4 pg 176

• Graph 3x+y=2

• Solve for y

• y=-3x+2

• Make our table

• If x= -2 -1 0 1 2

Then

• y=-3x+2= 8 5 2 -1 -4

Then graph them…

• Dot the dots

• Connect thedots

**Ex. 59-62**

Example 5 pg 177 The vertical line

• Take x+0*y=3

• Then x=3

• For x=3 y can equal ALL NUMBERS

**Ex. 63-74**

Example 5 pg 178

• Adjusting the scale… what if –5 to 5 isn’t enough on the graph labels?

• y=20x+500

• If x = -20 -10 0 10 20

Then

• y=20x+500 100 300 500 700 900

Plottin’

• The xand y axesCAN bedifferentif it’s needed

• Still evenspacing!

**Ex. 75-80**

Now to the short cut

• Isn’t making that table of plotted points a bit time consuming (read: boring?)

• How about a shortcut?!

Definitions

• Since 2 points make a line, let’s just find two easy to find points, graph then and connect them.

• Job done

• We’ll find the x-intercept (where the graph crosses X)and the y-intercept (where it crosses Y)

Example 7 pg 178 – How it’s done

• Graph 2x–3y=6

• To find the x-intercept we MAKE y=0 (since that is what y equals on the x line)

• 2x-3*0=6

2x=6

x=3 So our first point is (3,0)

Example 7 pg 178 y-intercept

• Next plug x=0 into 2x-3y=6 to find the y-intercept (since x=0 on the y line everywhere)

• 2*0-3y=6

-3y=6

y=-2 so our next point is (0,-2)

Plot the two points and draw!

• Andputtheequationinforothers

**Ex. 81-88**

Example 8 pg 179

• House Plan Cost C=900+30x

x= the number of copiesa) Cost for 8? C=900+30(8) = $1140

b) Intercepts: x=0 C=900+30(0) = $900

(even with no copies, there is a base price.) (0,900) and the other is (-30,0)

c) Then graph it…

Example 8 continued.

• c) graph it:

• d) interpret:

As the number of

copies drops, thecost drops…

**Ex. 89-92**

Application example 9 pg 179

• We are given the equation that describes the demand for tickets for the Ice Gators hockey game

• d=8000-100p

• Where d= the number of tickets sold, and p is the price in dollars

Example 9 continued

• a) How many tickets will be sold at $20 per ticket?

• Plug 20 into the equation for p=price

• d=2000-100*20=6000

• So at $20, we’d expect 6000 tickets

Ex 9b• Find the intercepts and interpret them• Replace d=0 in d=8000-100p

0=8000-100p

100p=8000

d=80 • Replace p=0 in d=8000-100p

d=8000

We have (p,d) and (0,8000) and (80,0)

If the price is free ($0), we’ll sell 8000 tickets (all)

If w up the price to $80, we’ll not sell ANY tickets

Ex 9c Graph it

• d) What happens to the demand as price increases?

As price increases, demand goes DOWN.

Try it! Section 3.1 Problems

• Try your hand at graphs…• Definitions Q1-6• Find the solution of the pairs Q7-16• Missing coordinates Q17-22• Plot simple points Q23-50• Graph each equation plotting at least 5 points

Q51-88• Graph each (can use x and y intercepts) Q88-102

Section 3.2 Looking closer at the ‘m’ the SLOPE

• The slope and you– How fast are you climbing uphill?– For example, if you climb 6 feet for every 100

feet you drive forward, you are on a slope of 6/100 or 6%.

– It is how much you are climbing as you go out horizontally.

– It is how much you change in y as you go out in x.

Sloping the Definition

• SLOPE– Slope = the Change in the y-coordinate

the Change in the x-coordinate

Slope Examples• Rise over Run! change in y over change in x

• Slope = 2/3 Slope = -2/-3 = 2/3

same thing!

Example 1page 188

• a) m= -4/3

Ex 1b

• b) m=2/3

Ex 1c

• m=-2/-4 = ½

**Ex. 7-10**

Using ANY two points gives us the same slope! Ex 2 page 189

• We have points A, B,C. Let’s find the slope between different pairs.

• A&Bm=rise/run=1/4

• A&Cm=rise/run=2/8=1/4

• B&C m=rise/run=1/4

Now for a change in slope finding…

• Remember when we had answers to our y=mx+b equation like (2,5) and (3,8)?

• We can find the slope from those too!

Definition• The Coordinate Formula for Slope

(stick this in your memory!)

m= (y2-y1) This is just rise over run again.

(x2-x1)

Provided x2-x1 is not equal to zero.

Example 3a pg189

• a) Find the slope of the following:(0,5) and (6,3)

so (x1,y1) and (x2,y2)

m=(y2-y1)/(x2-x1) = (3-5)/(6-0) = -2/6 = -1/3What if we reversed the points?

m=(y2-y1)/(x2-x1) = (5-3)/(0-6) = 2/-6 = -1/3No difference. You can’t go wrong!

Example 3b

• b) Find the slope of the following:

(-3,4) and (-5,-2)

so (x1,y1) and (x2,y2)

m=(y2-y1)/(x2-x1) = (-2-4)/(-5-(-3)) = -6/-2 = 3

Example 3c

• c) Find the slope of the following:

(-4,2) and the origin = (0,0)

so (x1,y1) and (x2,y2)

m=(y2-y1)/(x2-x1) = (0-2)/(0-(-4)) = -2/4 = - ½

**Ex. 19-32**

Caution… point 2 minus point1don’t mix the directions

m= (y2-y1) This will give the wrong sign.

(x1-x2)

Example 4 page 190

What about vertical lines?• 4a) (2,1) and (2,-3)

• m=(-3-1)/(2-2) = -4/0

• Explosion!

• Undefined slope

• Infinite slope

Ex 4b

• (-2,2) and (4,2)

• m=(2-2)/(4-(-2))

m=0/6 = 0

Horizontal lines = zero

slope

**Ex. 33-38**

Checking sign at a glance• If you see the line going UP as you read left

to right, it is a positive slope.

• If you see the line going DOWN as you read left to right, it a negative sign.

Example 5 page 191– Graphing a line if you have the

slope and a point

• You are given a point (step 1- plot it!) then the slope (Step 2 :rise up then run out to the 2nd point, plot it!). Step 3- draw the line.

• Graph the line through (2,1) with slope 3/4

Graph the line through (2,1) with slope 3/4

b) Graph the line through (-2,4) with slope -3

• The slope of –3 is –3/1 (same thing- right?)

Parallel lines are lines with the same slope!

• Draw a line through (-2,1) with slope ½ AND another line through (3,0) with slope ½

• Same slope = parallel lines!

• Different anchor point!

Example 6 page 193

**Ex. 45-46**

What about Perpendicular lines?

• Definition:Perpendicular Lines

Two lines with slopes m1 and m2 are perpendicular ONLY if

m1= - 1/m2

Also, ANY vertical line is perpendicular to any horizontal line.

Restating it

Example 7 page 194

• Graphing Perpendicular lines…

• We’ll draw them through point (-1,2)

• Slope 1= m1= -1/3 Slope 2= m2= 3

The graph on the next frame says it all…

(A picture is worth a 103 words.)

Perpendicular Graph

**Ex. 47-54**

Example 8 page 194

Are two lines parallel, perpendicular, or neither?

a) Line 1 goes through (1,2) and (4,8)

Line 2 goes through (0,3) and (1,5)

Line 1 slope = 8-2/4-1 = 6/3 = 2

Line 2 slope = 5-3/1-0 = 2/1 = 2 parallel!

Example 8

b) Slope line 1: 7-5/3-(-2) = 2/5

Slope line 2: 9-4/6-8 = 5/-2 = -5/2 perpendicular

c) Slope line 1: 6-4/-1-0 = 2/-1 = -2

Slope line 2: 4-7/4-7 = -3/-3 = 1 neither

**Ex. 55-62**

Using it in real life… Ex9 page 195

• If a car goes from 60mph to 0mph in 120ft… find the slope of the line.

• m=(60-0)/(0-120)= -0.5

• What is the velocity at 80feet?

• In 80 feet it drops -.5(80) or -40mph; 60-40=20mph

Slope Games in 3.2

• Yellow lines have homework problems…

• Definitions Q1-6

• Find slops from plotted lines Q7-18

• Calculate the slope from points Q19-38

• Graph lines through a point with slope Q39-44

• Parallel? Perpendicular? Neither? Q45-62

• Solve problem and graph Q63-72

Section 3.3 Putting y=mx+b together with your slope knowledge

• What if we don’t know a point, but know one point and the slope.

• We can play with the expression of slope to get y=mx+b

• y=mx+b is soooo important, you should end up dreaming about this equation

• m= the slope, b = the y-intercept (where x=0)!

Getting to y=mx+b

• Given one point (0,1) and the slope 2/3, we call the other point (x,y) since we don‘t know what it is.

• Remember slope? m=(y2-y1)/(x2-x1)

• Plug in the numbers and x and y above

• We get (y-1)/(x-0) = 2/3

• Rewrite it (y-1)/x = 2/3 why write a 0?

continuing

• Our last line was: (y-1)/x = 2/3• Now solve for y so we can get y=mx+b• Multiply both sides by x• y-1 = 2/3 x

Add 1 to both sides• y= 2/3 x + 1 DONE, now we can graph it.• We know that if x=0, y=1 so one point is (0,1)

[Hey! They told us that to begin with!] and m=2/3

Graphing it

Definition Again!• Slope- intercept form

y= mx + b

We know one point is always (0,b) and the slope m gives us all we need to make ANY line!

Example 1a pg 202

Using the slope-intercept form• Write an

equation from: y-intercept=(0,-2) so b=-2

• m=3/1=3

• y=3x-2

Example 1b

• Write an equation from: y-intercept=(0,0) so b=0

• m=2/2=1• y=1x-0 y=x

Example 1c

• Write an equation from: y-intercept=(0,5) so b=5

• m=-2/3=-2/3• y=-2/3x+5 y=x**Ex. 7-18**

Making it y=mx+b

• Example 2 page 203• What if you have a goofy starting equation?• Make it y=mx+b !• 3x-2y=6 Solve for y!• -2x=-3x+6 subtract 3x from both sides• y=-3/-2 x +6/-2 divide by –2 on both sides• y= 3/2 x – 3 simplify• RIGHT OFF: m=3/2 and b= -3 or (0,-3)**Ex. 19-38**

The Standard Form

• Every line EVER created is found in the standard form.

• Ax+By=C has every line in it!

• You just need to solve for y to find the y=mx+b form.

An example (3/page 203 ) of going backwards TO the standard form

• Starting with y=2/5 x + 3 we want Ax+By=C

• We are solving for the constant, 3! Weird!

• -2/5x +y = 3 I added –2/5 from both sides

• Done! But we could get rid of the fraction

• 5(-2/5 x) + 5y = 5*3 multiply everything by 5

• -2x + 5y = 15 So A=-2, B=5, C=15

**Ex. 39-54** (no homework)

Using y=mx+b to make a graph

• So you can be given any linear equation with two variables and graph it!

Example 4 page 204

• Graph 2x-3y=3• Solve for y• -3y=-2x+3 subtract 2x • y=-2/-3 x + 3/-3 divide by -3• y= 2/3 x –1 clean it up• And you can see the m=2/3 and b=-1 which is

the point (0,-1)

**Ex. 55-56**

Ex 4 the graph

Another example… Ex 5a page 204

• Graph y=-3x+4

• No solving needed

• m=-3

• b= +4

or (0,4)

Ex 5b

• 2y-5x=0

• Add 5x to bothsides

• 2y=5x+0

• Divide by 2

• y=5/2 x + 0

**Ex. 57-68**

Goofing around with section 4.3page 205

Example 6aA line through (0,3) parallel to y=2x-1

• We know the slope is 2. Done.

• From the ()’s the y-intercept is 3. Done.

• Just write the equation…

• y=mx+b y=2x+3

Example 6b• We want to write the y=mx+b (the slope intercept

form) of a line through (0,4) (hey that’s the b!) PERPENDICULAR to 2x-4y=1

• One step at a time… what is the y=mx+b form of the ‘other’ line?

• -4y=-2x+1 y=-2/-4 x – ¼ y= 1/2x- ¼ • So the perpendicular slope to this is –1/m • -1/(1/2) = -2 so this is the slope we want!• We’re done! m=-2 and b=4 (given right out)• y= -2x +4 **Ex. 77-90**

Ex 7 page 206 Application• If a landscaper has $800 to spend on bushes

which are $20 each and trees at $50 each. If x is the number of bushes and y is the number of trees then 20x+50y=800. Write it in slope-intercept form (y=mx+b)

• 20x+50y=800 50y = -20x +800 subtract 20x from both

sides

y= -20/50 x + 800/50 divide by 50

y= -2/5 x + 16

done!

Exercises 3.3Practice makes sore hands

• Work with y=mx+b!• Definitions Q 1-6• Write the equation from the graph Q7-18• Find the slope and y-intercept Q19-38• Write equations in Ax+By=C Q39-54• Draw the graphs Q55-68• Parallel or Perpendicular? Q69-76• Write the y=mx+b form Q77-90• Verbal problems Q91-104

Section 3.4 The point slope form

• It’s STILL y=mx+b but we’re peering into the ‘box’ a bit more

• If you have a graph, can you write y=mx+b?

• SURE!

Definition

• And doing this is using the point-slope form of y=mx+b

For example

• If you have some line through (4,1) and know the slope is 2/3

• (shown here)

Example continued• We know the slope is (y2-y1)/(x2-x1) = m• Plug in what we know…• (y2-1)/(x2-4) = 2/3 • (y-1)/(x-4) = 2/3 drop the subscripts- who

needs ‘em?• y-1 = 2/3(x-4) multiply both sides by (x-4)• Stop there! (we could go to y=mx+b but not this

time). It looks like we want it (below).

Why are we stopping there?

• Why not take it to y=mx+b ?

• Because it makes problems (common) where you know the slope and ANY point really easy!!!

• (With y=mx+b you know the slope and JUST the y-intercept point)

• Let’s see how nice it is…

Example 1 page 213

• Find the equation through (-2,3) with slope ½ . • (-2,3) is NOT the intercept, it’s just some point

• Use the point-slope form y-y1=m(x-x1)

• y-3 = ½ [x-(-2)] plug in (-2,3) and ½• y-3 = ½ (x+2) simplify• y-3 = ½ x +1 simplify• y= ½ x +4 add 3 to both sides

Example 1 the old way

• If we started with the slope-intercept form (not the point-slope form) we’d do it this way…

• y=mx+b plug in ½ for m and (-2,3) for (x,y)• 3= ½ (-2) + b simplify• 3= -1 +b add 1 to both sides• 4=b • Then write it out, m=1/2 (given) b=4• y= ½ x + 4**Ex. 7-24**

Example 2 page 214 With two points no slope

• We don’t know the slope like before, but with two points (-3,-2) and (4,-1) we can get it…

• m= (-2-(-1))/(-3-4) = -1/-7 = 1/7

• Now we are ready. With the slope and ONE point we can use the point-slope form

• next frame…

Ex 2 continued• We found m= 1/7 and we can use either point – why not (-

3,-2) (feeling negative tonight?)

• y-y1 = m(x-x1) plug in the numbers

• y-(-2) = 1/7 [x-(-3)] simplify• y+2 = 1/7(x+3) multiply by 7 to get rid of 1/7th • 7(y+2) = 7[1/7(x+3)] simplify• 7y+14 = x+3 subtract 14 from both sides• 7y = x-11 We want Ax+By=C = standard form• -x+7y = -11 -1(-x+7y)= -1(-11) multiply by -1• x-7y = 11 done! **Ex. 25-44**

Parallel Lines - revisited• Example 3 page 215

Write the equation of the line parallel to the line y=-3x+9 and contains (2,-1). Give the answer in slope-intercept form (y=mx+b)

• First write 3x+y=9 in slope intercept form• y=-3x+9 (only had to subtract 3x from both sides!)• So both lines have slope m=-3

• Now we’re back to having a slope (-3) and finding a line through a given point (2,-1) . You can forget the word parallel if it makes you start to sweat.

Ex 3 continued

• m= -3 and point (2,-1)

• Use the point-slope form… it’s what we have!

• y-y1=m(x-x1)

• y-(-1)=-3(x-2) plug in the numbers

• y+1 = -3x +6 simplify

• y= -3x + 5 subtract by –1 , done!

Ex 3 visually

**Ex. 49-50**

Perpendicular Lines - revisited

• Again, this is just a smoke and mirrors way to give us our slope. m1 = -(1/m2)

• Example 4 page 216 Find an equation perpendicular to 3x+2y=8 and contains (1,3).

• The first part gives us the slope we can flip and make negative. The second part will help us with the point-slope equation… here we go…

Example 4 page 216

• 3x+2y=8 make it into y=mx+b so we can get m

• 2y = -3x + 8

• y= (-3/2) x + 4 can you see the steps?

• So m = -3/2 our perpendicular line will have

m= -(1/(-3/2)) = 2/3 see it?

Hey! We have a slope m=2/3 and a point (1,-3)

Let’s use y-y1=m(x-x1) again!

Example 4 continued• y-y1=m(x-x1) m=2/3 through point (1,-3)

• y-(-3)=2/3(x-1)• y+3 = 2/3 x – 2/3• y= 2/3 x – 2/3 – 3 • y= 2/3 x –2/3 – 9/3• y= 2/3 x –11/3 Done!

**Ex. 51-60**

Section 3.4 Work it out!

• Definitions Q1-6

• Write stuff in slope intercept form Q7-14

• Equation of a line with point and slope Q15-24

• Write only integers Q25-30

• Equation of a lien with two points given Q31-44

• Parallel and Perpendicular Q45-48

• Find equation of a new line Q49-60

• Mix bag Q61-78

• Application Problems Q79-100

Section 3.5 Variation

• A new section that lets you ‘see’ what is happening with this linear equations.

• They MEAN something in the REAL WORLD… here is a chance to see what that is…

Things that Vary

• One thing can vary when another thing does in the same way it does (direct), the opposite that it does (inverse), or generally along with a few other things (jointly).

The Direct Variation

• An example: D=60T

T (hours)

1 2 3 4 5 6

D (miles)

60 120 180 240 300 360

Directly direct?

‘y varies directly as x’ or

‘y is directly proportional to x’

Always looks like: y=kx

k is just a constant that isn’t zero

Inverse Variation

An Example: T= 400/R

As R gets big, T gets small.

R (mph)

10 20 40 50 80 100

T (hours)

40 20 10 8 5 4

Inversely!

‘y varies inversely as x’ or

‘y is inversely proportional to x’

y= k/x

k is a nonzero constant

Joint Variations(no cigarettes!)

‘y varies jointly as x and z’

or ‘y is jointly proportional to x and z’

y=kxz

k is a nonzero constant

Example 1 page 226

• Writing it out!

a) a varies directly as t a=kt

b) c is inversely proportional to m c=k/m

c) q varies jointly as x and y q=kxy

**Ex. 5-14**

Example 2 page 226

Finding the constanta)

a varies directly as x and a=10 when x=2

We start with a=kx plug in the numbers:

10=k2 or 10=2k (divide by 2) k=5

The formula is a=5x

Example 2b

• w is inversely proportional to t and w=10 when t=5

w=k/t plug in: 10=k/5

Multiply by 5

50=k so w=50/t

Example 2c

• m varies jointly as a and b, and m=24 when a=2 and b=3

m=kab plug in: 24=k(2)(3) 24=6k

Divide by 6 k=4

So m=4ab done! **Ex. 15-24**

Applications: Example 3- Direct Variation

• Read the problem on page 227

The amount A of the electric bill varies directly as the amount of E of electricity used.

A=kE for some constant k

Since E=2800 and A=196 196=2800k

Which gives us k=0.07

Using the equation and if E=4000 we get:

A=0.07(4000)=280 so the bill would be $280

**Ex. 25-26**

Inverse Example 4 page 227

• Because the volume V is inversely proportinal to the pressure P we have

V=k/P for some kFind k! V=12 when P=20012=k/200 so k=2400If P=150 we can use V=2400/PV=2400/150 = 16, so the volume is 16 cm3

when the pressure is 150 kg/cm2

Joint variation problem Ex 5 page 228

• Because C varies jointly with the weight and distance (w and d) we have:

C=kwdFind k! C=3000, W=2500, d=6003000=k(2500)(600) use your calculator!k= 0.002So now w=1500 and d=800 and we found

C=0.002wd C=0.002(1500)(800) = $2400

Section 3.5 problems

• Definitions Q1-4

• Direct, Inverse, and Joints Q5-14

• Finding that k thing Q15-24

• Applications Q25-56

Section 3.6One small step in graphing

• Now we’ll make an unholy marriage between the inequalities we worked on in Chapter 3 and the graphing we did earlier

• Was it EVER INTEDED to happen?

• Well, ok, sure it was.

Section 3.6 Definition

• Linear Inequalities in Two Variable• If A,B, and C are real numbers (A&B both

can’t be zero) then,Ax + By < C

Is called a linear inequality in two variables.

We can put <=, >, or >= in place of < above as well.

The family album of linear inequalities on two variables

• 3x-4y<=8

• y>2x-3

• x-y + 9 < 0

• x+5y <= 22

• Etc.

The True or False Game Returns

• Does it work?

• Example 1 pg 232 With 2x-3y >= 6

• a) Try (4,1) remember it is (x,y) always

2(4) – 3(1) >=? 6

8-1 >=?6

7 >=?6 NO!

Ex 1b & c2x-3y >= 6

b) Try (3,0) 2(3)-3(0) >=? 6 6 >=? 6 YES!

c) Try (3,-2) 2(3) –3(-2) >=? 6

6 +6 >=?6 12 >=? 6 YES! **Ex. 7-14**

Playing the field - again

• We just tested it point by point. That could take all day… or all year… or the rest of time to test EVERY possible point.

• Why not just draw the boundary line and shade in where it is true?

• We’ll map out the entire field and have the rest of our lives to do something else!

If you have y>x+2 what happens?• So this is all places on the graph to one side or the other of

y=x+2 where y is larger than the line.• For

Example(3,5) is on the line

• (3,6) isabove itand true

• Note the dashed linemeans it is not >=

• We’d draw it solid for>=

The cookbook

Example 2 Doing it! Page 234

• Graph thema) y < 1/3 x + 1The slope is m=1/3The intercept is (0,1)y is “less than”blaa blaa so shade“below”• Dash the line

because of <

Example 2b

• b) y>=-2x +3• m=-2• Thru pt. (0,3)• Less than

means shadebelow

• Equal sign means solidline not dashed

Example 2c• c) 2x-3y<6

Solve for y• -3y<-2x+6

y> 2/3x –2• m=2/3• Pt (0,-2)• y ‘is greater

than’ so shadeabove

• Dashed line from <

**Ex. 15-28**

Special Cases: horizontal and vertical lines (Ex 3)

• a) y <= 4

y=4 is a horizontal

line through (0,4)

[there is just no x]

• Less than meansshade below

Ex 3b

• x>3 The line is allx=3 points• > means dashed

line• > means ‘to

the right **Ex. 29-32**

Confused about shading?

• Why not pick a test point after drawing the line (dashed or solid)?

• (You can do this until you are confident in which side the shading goes on for > or < inequalities).

• Remember ! Always make it look like y > mx+b or

y < mx+b or y>=mx+b or y<=mx+b

Example 4 pg 236

The Test Point Shading Trick• Graph the inequality 2x-3y>6• Solve for y• -3y>-2x+6• y< (-2/-3)x –6/2• y< 2/3 x –3• m=3/2, y-int. (0,-3)

• We’ll test with (0,1).• If true, we’ll shade there!

Example 4 continued

• Plug out test point (0,1) into 2x-3y>62*0-3*1>6-3>6 FALSE

• So we must shade the OTHER side…

Example 4b continued

x-y<=0First graph x-y=0By getting it to y=-x

Select, say (1,3) and test:1-3<0The line is solid since itis <=

**Ex. 39-50**

Another application Ex 5 page 237

• The company can obtain at MOST 8000 board feet of oak. It takes 50 board feet for round tables, 80 board feet for rectangle tables. What are all the possible combinations of round and rectangle tables they can make?

• The sum of the number of both kinds of tables is less than or equal to the 8000 max.

• 50x+80y <= 8000

Ex 5 continued

• Find the intercepts (easier than solving for y and using the slope, since our units are strange and large – 100’s).

• The intercepts occur where first x=0, then y=0.

• x=0 50*0+80y=8000• 80y=8000• y=100 giving us (0,100)

Ex 5 goes on

• Then we find y=0

• 50x+80*0 =8000

• 50x=8000

• x=160 so we have (160,0)

Ex 5 graphed with (0,100) and (160,0)

**Ex. 51-54**

Exercises 3.6

• Definitions Q1-Q6• Which points satisfy? Q7-14• Graph the inequality Q15-50• Word Problems Q51-56• Definitions Q1-6• Which points solve both inequalities? Q7-12• Graph them! Q13-44• Application problems Q45-47